Opportunities to improve climate models lie in exploiting a larger
fraction of the available observational data together with high-resolution simulations, and learning from both, systematically and not manually [1]}. To fully account for parametric uncertainty, we adopt a Bayesian view of the model-data relationship, which amounts to solving a Bayesian inverse problem. Given model, data, and prior information on the parameters, Bayesian inversion yields a posterior distribution of parameters. The mean or mode of the posterior distribution define the best parameter estimate, and the entire distribution provides uncertainty quantification through its spread about the mean or mode. We use the mismatch between climate statistics simulated with the model and those obtained from observations or high-resolution simulations as the data likelihood in the Bayesian inverse problem to calibrate parameterizations in a climate model and to quantify their parametric uncertainties. Our focus is on learning from time-averaged climate statistics for three reasons: (1) time-averaged statistics are what is relevant for climate predictions; (2) time-averaged statistics vary more smoothly in space than atmospheric states, leading to a smoother optimization problem than that of atmospheric state estimation in numerical weather prediction (NWP); (3) time-averaging over long time-intervals reduces the effect of the unknown initial state of the system,
removing the need to determine it. Focusing on time-averaged climate statistics, rather than on instantaneous states or trajectories as in NWP, makes it possible to exploit climate observations and high-resolution simulations even when their native resolutions are very different from those of climate models.
